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Quadratic Formula Calculator

Solve ax² + bx + c = 0 step by step: discriminant, nature of roots, real or complex roots and vertex — free quadratic formula calculator for Algebra, SAT, GCSE and JEE.

2x2−4x−6=02x^{2} - 4x - 6 = 0

Roots

x1=3,x2=−1x_1 = 3, \quad x_2 = -1
2x2−4x−6=2(x−3)(x+1)2x^{2} - 4x - 6 = 2(x - 3)(x + 1)
  1. 1

    Identify a, b and c

    Compare with the standard form ax² + bx + c = 0.

    2x2−4x−6=0⇒a=2,  b=−4,  c=−62x^{2} - 4x - 6 = 0 \quad\Rightarrow\quad a = 2,\; b = -4,\; c = -6
  2. 2

    Compute the discriminant

    D = b² − 4ac decides how many roots there are and what kind.

    D=b2−4ac=(−4)2−4⋅2⋅(−6)=16+48=64D = b^{2} - 4ac = (-4)^{2} - 4\cdot2\cdot(-6) = 16 + 48 = 64
  3. 3

    Nature of the roots

    D > 0 and 64 is a perfect square, so there are two distinct real rational roots.

  4. 4

    Apply the quadratic formula

    x=−b±D2a=4±644x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{64}}{4}
  5. 5

    Simplify the square root

    64=8⇒x=4±84\sqrt{64} = 8 \quad\Rightarrow\quad x = \frac{4 \pm 8}{4}
  6. 6

    Split into the two roots

    x1=4+84=3,x2=4−84=−1x_1 = \frac{4 + 8}{4} = 3, \qquad x_2 = \frac{4 - 8}{4} = -1
  7. 7

    Vertex of the parabola

    h = −b/2a and k = c − b²/4a = −D/4a. The parabola opens upward (a > 0), so the vertex is a minimum; the axis of symmetry is x = h.

    h=44=1,k=−648=−8⇒(h,k)=(1, −8)h = \frac{4}{4} = 1, \qquad k = \frac{-64}{8} = -8 \quad\Rightarrow\quad (h, k) = \left(1,\ -8\right)
  8. 8

    Factored form

    Rational roots r₁, r₂ give a(x − r₁)(x − r₂); fractions are cleared into the brackets.

    2x2−4x−6=2(x−3)(x+1)2x^{2} - 4x - 6 = 2(x - 3)(x + 1)

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FAQ

What is the quadratic formula?

For ax² + bx + c = 0 the roots are x = (−b ± √(b² − 4ac)) / 2a.

What does the discriminant tell you?

D = b² − 4ac. D > 0: two real roots, D = 0: one repeated root, D < 0: two complex roots.

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